Fitting linear orthogonal polynomials
A linear polynomial assumes that three or more groups have an ordering e.g. A>B>C and that the distances between each successive pair of groups are equal. This may be tested by specifying contrast coefficients in an anova.
The contrast coefficients may be obtained in R using contr.poly(p) for a polynomial of degree p. Tables of polynomial contrast coefficients are also available in Biometrika tables which also give the formulae to obtain integer contrast coefficients.
contr.poly(4,contrasts=TRUE)
Specific contrast coefficients can also be obtained assuming unequal distances between groups but the ratio of these differences must be specified e.g. if groups B and C are twice as far from each other as groups A and B then contrast coefficients may be obtained using, for example, R.
> x <- c(1,2,4) > poly(x,degree,1)
If you suspect the groups are unequally spaced but do not know by how much then either fit a quadratic polynomial in addition to a linear one using a second set of contrast coefficients or use a nonparametric procedure such as Jonckheere's Trend Test.
Reference
Biometrika tables for Statisticians Volume 1 (1954) Edited by Pearson, ES and Hartley, HO. Cambridge University Press.